A193288 E.g.f.: A(x) = 1/(1 - 3*x^2)^(1 + 1/(3*x)).

1, 1, 7, 28, 289, 2131, 29161, 316072, 5395993, 77326165, 1583326171, 28229026156, 674412621697, 14384156661343, 392879390385301, 9753823992141496, 299849358712509361, 8492478062686906057, 290226665437376352463, 9233909417529486840412

Offset: 0

Author

Paul D. Hanna, Sep 07 2011

Keywords

• nonn

Comments

More generally, we have the identity:

Sum_{n>=0} (x^n/n!)*Product_{k=1..n} (1+k*y) = 1/(1 - x*y)^(1 + 1/y); here y=3*x.

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 28*x^3/3! + 289*x^4/4! + 2131*x^5/5! +...
where A(x) satisfies:
A(x)^(3*x/(1+3*x)) = 1 + 3*x^2 + 9*x^4 + 27*x^6 + 81*x^8 + 243*x^10 +...
Also,
A(x) = 1 + x*(1+3*x) + x^2*(1+3*x)*(1+6*x)/2! + x^3*(1+3*x)*(1+6*x)*(1+9*x)/3! + x^4*(1+3*x)*(1+6*x)*(1+9*x)*(1+12*x)/4! +...
The logarithm begins:
log(A(x)) = x + 3*x^2 + 3*x^3/2 + 9*x^4/2 + 9*x^5/3 + 27*x^6/3 + 27*x^7/4 +...
a(0) = 1; a(n) = (n-1)! * Sum_{k=1..n} k * 3^floor(k/2)/floor((k+1)/2) * a(n-k)/(n-k)!. - _Seiichi Manyama_, Apr 30 2022
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..412

Crossrefs

Cf. A193281, A193287, A193289, A193290.

Programs

Mathematica

CoefficientList[Series[1/(1-3*x^2)^(1+1/(3*x)), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 25 2013 *)

PARI

{a(n)=n!*polcoeff(1/(1 - 3*x^2 +x^2*O(x^n))^((1+3*x)/(3*x)),n)}

PARI

{a(n)=n!*polcoeff(sum(m=0,n,x^m/m!*prod(k=1,m,1+3*k*x+x*O(x^n))),n)}

PARI

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=1, i, j*3^(j\2)/((j+1)\2)*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Apr 30 2022

Formula

E.g.f.: A(x) = Sum_{n>=0} x^n/n! * Product_{k=1..n} (1 + 3*k*x).

a(n) ~ n! * n^(1/sqrt(3))*3^(n/2+1/2)/(2^(1+1/sqrt(3))*Gamma(1/sqrt(3))). - Vaclav Kotesovec, Jun 25 2013